where u is a numerical factor, we have zeroes of illumination for the values

a = 1-220, 2-233, 3-238, 4-241, 5-243, . . .

and within these rings respectively is comprised a fraction of the total illumination amounting to

•839, -071, -028, -015, -009, . . .

while the maximum brightness occurs for the values

u = -000, 1-63,5, 2-679, 3-699, 4-710, 5,717, . . .

and at these points the relative intensities are

1-0000, -0175, -0042, -0016, -0008, -0004, . . .

The only telescopic feature permitting control of this phenomenon is the aperture. If this is increased the angular diameters of the rings are all diminished in proportion. It is of interest to note that if the objective is reduced to an annulus by a central stop, the rings are somewhat diminished in diameter, but at the expense of much less favourable distribution of the proportions of tho light between them, the shares of the outer rings being increased. Numerically, considering, say, a 10-inch object glass and the D-ray, the diameter of the spurious disc of a star lying within the first dark ring would be 0"-98. Hence the* images of the members of a close pair would be more or less superposed if their distance was less than 1"; but much within this a good eye would detect the elongation of the united image, aa indeed- was found in practice by S. W. Burnham, who discovered when using a 6-inch objective by Alvan Clark many double stars the separation of which lay beyond this theoretical " resolving power."

Owing to the shorter wave-length of actinic rays it might seem that photography was better circumstanced than the eye for receiving small images, but it is well known that this is not the case. From a variety of reasons, the photographic image spreads ; and indeed the diameters can be used as measures of the star's magnitude. If the smallest recorded images are of less diameter than 2" or 3", the results would as a rule bo considered very favourable.

§ (8) OPTICAL GLASS. — So far we have spoken of the geometrical modification of the beam of light; we shall now consider briefly the characteristics of the glasses1 out of which the lenses are constructed. Apart from the limitations which technical difficulties of glass manufacture impose, geometrical discussions are merely mathematical exercises. The earliest telescopes were made with a single lens as objective, and in consequence suffered severely from dispersion, the images for different colours being ranged along the axis. The focal length was made as groat as possible,

See also article " Glass.'

and, for example, D. G. Cassini discovered two of the satellites of Saturn, Tethys and Dione, with telescopes of 100 ft. and 136 ft. in length respectively. But since the separation as well

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VOL. IV

PIG. 14.

as the size of the images corresponding to, say, the rays C, F (Fig. 14) would increase proportionately with the focal length, it is an error to suppose that increase of focal length waa any cure for chromatic faults, except in so far as the narrow beam does not allow spreading of any description. Achromatism was discovered and the principles published by John Dollond about 1760,2 though achromatic lenses had been made before him by Chester Moor Hall, a private gentleman, as early as 1733. Making the converging lens of crown glass, the principle was to annul wholly the dispersion while only partly annulling the deviation by associating with the crown lens a divergent lens of greater proportionate dispersive power. This was found in " flint" glass or " English crystal," a dense and brilliant glass, which contained as base a silicate of lead. To obtain large homogeneous pieces of flint glass, free from striae, for negative lenses, proved a very difficult technical problem. It was solved by P. L. Guinaud of Neuchatel, about 1800, whose secret was to stir the pot of melted glass with a fireclay rod to the last possible moment, and, allowing it to cool by itself, to take its natural fractures as .marking off lumps of homogeneous constitution. When a large one was found this was softened again and moulded into tho form of a disc. This secret he taught to Fraunhofer's firm in Bavaria and to Fraunhofer's successors, Merz and Mahler, by whom it was jealously guarded, so that in the middle of the nineteenth century they were the only people who could produce an objective of even so moderate a size as 8 inches diameter. Guinand, however, returned to Neuchatel, and his method passed to his son H. Guinand, and in succession to Foil, Mantois, and Parra in Paris, and through one of H. Guinand's collaborators, G. Bon-temps, who took refuge in this country in the political disturbances of 1848, to Messrs. Chance in Birmingham. All the greatest lenses in the world, so far produced, excepting the 32-inch at Potsdam,8 came from one or other of these firms. Their productions were, however, conservative, and a separate rcvolu- • tion in glass-making came from the researches

» Phil. Trans., 1758, p. 733. a Even these glasses, by Schott & Co,, are of tho old types.

3i